Asymptotic geometry of the moduli space of parabolic SL(2,C)-Higgs bundles - Fredrickson, Laura; Mazzeo, Rafe; Swoboda, Jan; Weiß, Hartmut (Journal of the London Mathematical Society, 2022)
Given a generic stable strongly parabolic SL(2,C)-Higgs bundle (E,φ), we describe the family of harmonic metrics ht for the ray of Higgs bundles (E,tφ)t≥ 0 by perturbing from an explicitly constructed family of approximate solutions htapp. We then describe the natural hyperkähler metric on M$\mathcal M$ by comparing it to a simpler ‘semiflat’ hyperkähler metric. We prove that gL2−gsf=O(e−γt) along a generic ray, proving a version of Gaiotto-Moore-Neitzke's conjecture. Our results extend to weakly parabolic SL(2,C)-Higgs bundles as well. A centerpiece of this paper is our explicit description of the moduli space and its L2 metric for the case of the four-punctured sphere. We prove that the hyperkähler manifold in this case is a gravitational instanton of type ALG and that its rate of exponential decay to the semiflat metric is the conjectured optimal one, γ=4L, where L is the length of the shortest geodesic on the base curve measured in the singular flat metric |detφ|.